ABSTRACT
This chapter builds the mathematical structures up from the original foundation of set theory and logic. In particular, it discusses some major concepts in mathematics: relations, equivalence, and functions, and also discusses some logical tools. In mathematics, one often wants to compare (or relate) two objects together. Relations are not necessarily symmetric. This is even more evident when one notices that a relation is a collection of ordered pairs, and so clearly the order in which one lists the sets will matter. Relations can even be defined on sets other than sets of numbers. It turns out that the notion of an equivalence relation generalizes the usual notion of equality. Functions are a cornerstone of mathematics. The chapter describes several different desirable features that functions might possess. A function will not be surjective if there are points in the codomain that do not get mapped to.
